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The Millennium Problems
In 2000, the Clay Foundation of Cambridge, Massachusetts, announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive 1 million in prize money. There was some precedent for doing this: in 1900 David Hilbert, one of the greatest mathematicians of his day, proposed twenty-three problems, now known as the Hilbert Problems, that set much of the agenda for mathematics in the twentieth century. The Millennium Problems are likely to acquire similar stature, and their solution (or lack of one) is likely to play a strong role in determining the course of mathematics in the current century. Keith Devlin, renowned expositor of mathematics, tells here what the seven problems are, how they came about, and what they mean for math and science. These problems are the brass rings held out to today's mathematicians, glittering and just out of reach. In the hands of Keith Devlin, "the Math Guy" from NPR's "Weekend Edition," each Millennium Problem becomes a fascinating window onto the deepest and toughest questions in the field. For mathematicians, physicists, engineers, and everyone else with an interest in mathematics' cutting edge, The Millennium Problems is the definitive account of a subject that will have a very long shelf life.
237 pages, Paperback
First published January 1, 2002
23 people are currently reading
894 people want to read
About the author
Keith Devlin
85 books166 followersDr. Keith Devlin is a co-founder and Executive Director of the university's H-STAR institute, a Consulting Professor in the Department of Mathematics, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 26 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is "the Math Guy" on National Public Radio.
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Displaying 1 - 30 of 45 reviews
February 7, 2017
This book is an attempt to explain, at least where at all possible, the seven mathematical millennium problems, which the Clay Foundation in 2000 offered a one million dollar prize for the solving of each problem. They are in order of presentation in the book: The Riemann Hypothesis, Yangs-Mill Theory and the Mass Gap Hypothesis, The P vs. NP Problem, The Navier-Stokes Equation, The Poincare Conjecture, The Birch and Swinnerton-Dyer Conjecture, and The Hodge Conjecture.
The Riemann Hypothesis involves complex analysis and if proven true would solve the pattern of the primes. The Yang-Mills Theory and the Mass Gap Hypothesis involves quantum physics and its solution would be a big help to both mathematicians and physicists. The P vs. NP Problem involves computational complexity. If P can be shown to be the same as a certain class of NP, than it would be a big boost to solving complex problems by computer computation like the traveling salesman problem, which is the solution to the shortest route of an itinerary. If not, than we will have to settle for approximate solutions. The Navier-Stokes Equation involves fluid dynamics and if solved would help in things like airplane design. It would also have a bearing on Chaos Theory. The Poincare Conjecture involves topology, which studies what stays the same in a particular geometry after twisting or stretching, but not tearing. The last two are beyond even Keith Delvin’s ability to explain, as he fully admits, so I will not even attempt describing them.
The book was okay, and I did learn something about these difficult mathematical problems. Delvin is a good expositor for the knowledgeable reader. I may have lost some of my interest for mathematical books aimed at the general reader, except books dealing with infinity, which is why I said the book was just okay. However, I am still very attracted to the philosophy of mathematical books; these books are not in the general readership category.
My recommendation is for those who have an interest in mathematics, whatever your level of competency. If you know next to nothing, you will probably be lost most of the time; if you have some type of mathematical expertise, you should do just fine, I think, even in areas of mathematics were you might not know a whole lot. For those of you who are somewhere in between, you will probably get lost at times like I did.
The Riemann Hypothesis involves complex analysis and if proven true would solve the pattern of the primes. The Yang-Mills Theory and the Mass Gap Hypothesis involves quantum physics and its solution would be a big help to both mathematicians and physicists. The P vs. NP Problem involves computational complexity. If P can be shown to be the same as a certain class of NP, than it would be a big boost to solving complex problems by computer computation like the traveling salesman problem, which is the solution to the shortest route of an itinerary. If not, than we will have to settle for approximate solutions. The Navier-Stokes Equation involves fluid dynamics and if solved would help in things like airplane design. It would also have a bearing on Chaos Theory. The Poincare Conjecture involves topology, which studies what stays the same in a particular geometry after twisting or stretching, but not tearing. The last two are beyond even Keith Delvin’s ability to explain, as he fully admits, so I will not even attempt describing them.
The book was okay, and I did learn something about these difficult mathematical problems. Delvin is a good expositor for the knowledgeable reader. I may have lost some of my interest for mathematical books aimed at the general reader, except books dealing with infinity, which is why I said the book was just okay. However, I am still very attracted to the philosophy of mathematical books; these books are not in the general readership category.
My recommendation is for those who have an interest in mathematics, whatever your level of competency. If you know next to nothing, you will probably be lost most of the time; if you have some type of mathematical expertise, you should do just fine, I think, even in areas of mathematics were you might not know a whole lot. For those of you who are somewhere in between, you will probably get lost at times like I did.
July 20, 2014
I read this book long ago. This is probably as close as one can get to give a light overiview of the seven problems recognised by the clay institute for a million dollar prize. The author here takes up an impossible task of explaining these problems to a lay audience. Even if he didn't entirely succeed in this, this book can be used to spark someone's interest for deeper study. Worth the read at least for the chapters on Riemann hypothesis and the P vs NP problem.
September 10, 2011
A very inspirational book. Now I know what math problems I should start solving.
November 28, 2021
P versus NP
The Hodge Conjecture
The Poincaré Conjecture
The Riemann Hypothesis
Yang-Mills Existence and Mass Gap
Navier-Stokes Existence and Smoothness
The Birch and Swinnerton-Dyer Conjecture
So goes the list of the Clay Institute's Millennium Prize problems - the greatest unsolved problems in mathematics - each carrying a $1 million prize. The full list in this format is how the Clay Institute presented them on their poster, such that when they are centre aligned they form a Christmas Tree shape. Author Keith Devlin's masterstroke was to ignore the official 'ordering' of the problems and carefully choose how to go about describing these problems to the lay reader.
By starting with Riemann, Devlin lays the groundwork of complex numbers and calculus that will come up again and again in later chapters (such as Navier-Stokes). The following three chapters - Yang-Mills, P versus NP and Naiver-Stokes - all emerge in the applied sciences and, while very much questions of pure mathematics, nevertheless provide ample opportunity for the author to utilise their physical or computational origins to lead the reader gently up to the daunting problems themselves.
This all means that the reader is four chapters in before the really heavy stuff starts. Wisely, Devlin spends more time on these 'easier' (to understand, not necessarily to solve) problems. The fifth chapter on Poincaré has a lot of graspable elements that still keep it just within reach of the lay reader (topology affords lots of nice visualisations and intuitive explanations). Of course, as this book was written just after the problems were announced, it doesn't contain any details of the fact that Poincaré has actually been proved (there are only six outstanding Millennium Problems as of 2021).
Only by chapters six and seven will even those with a decent grasp of mathematics start to come unstuck. Birch and Swinnerton-Dyer, a problem about rational solutions of elliptic curves, and the Hodge Conjecture, which I won't even attempt a precis of, are so couched in layer upon layer of abstract mathematics that the author deserves praise for even attempting a lay explanation.
A very committed reader will get something out of these chapters - but I think the author was right to hold up his hands in a couple of places and say: this might just be too hard for you. He gives an opt-out about halfway through the Birch and Swinnerton-Dyer chapter; while he advises the reader not to feel bad if they have to bail out of the whole of the Hodge Conjecture chapter. Which is not to criticise the author for lack of trying. Rather, I think he does well on two counts. One: he did manage to give me some level of understanding of these highly esoteric problems. Two: he makes clear that these are incredibly advanced and, at least in the case of Hodge, he makes clear that mathematicians are not even in agreement as to what it means.
All in all, the author deftly leads the engaged amateur through a lot of extremely difficult problems, to such an extent that I've already made use of my understanding of the P vs NP problem in my work (I occasionally have to think about complexity theory). At the same time, he doesn't shy away from the very hardest topics but also shields the reader from the very worst of it and attempts to keep up their confidence with welcome honesty about his own struggles with the mathematics of the last few chapters. Overall, I doubt a better primer on the Millennium Problems exists and am thoroughly glad I finally got around to reading this.
The Hodge Conjecture
The Poincaré Conjecture
The Riemann Hypothesis
Yang-Mills Existence and Mass Gap
Navier-Stokes Existence and Smoothness
The Birch and Swinnerton-Dyer Conjecture
So goes the list of the Clay Institute's Millennium Prize problems - the greatest unsolved problems in mathematics - each carrying a $1 million prize. The full list in this format is how the Clay Institute presented them on their poster, such that when they are centre aligned they form a Christmas Tree shape. Author Keith Devlin's masterstroke was to ignore the official 'ordering' of the problems and carefully choose how to go about describing these problems to the lay reader.
By starting with Riemann, Devlin lays the groundwork of complex numbers and calculus that will come up again and again in later chapters (such as Navier-Stokes). The following three chapters - Yang-Mills, P versus NP and Naiver-Stokes - all emerge in the applied sciences and, while very much questions of pure mathematics, nevertheless provide ample opportunity for the author to utilise their physical or computational origins to lead the reader gently up to the daunting problems themselves.
This all means that the reader is four chapters in before the really heavy stuff starts. Wisely, Devlin spends more time on these 'easier' (to understand, not necessarily to solve) problems. The fifth chapter on Poincaré has a lot of graspable elements that still keep it just within reach of the lay reader (topology affords lots of nice visualisations and intuitive explanations). Of course, as this book was written just after the problems were announced, it doesn't contain any details of the fact that Poincaré has actually been proved (there are only six outstanding Millennium Problems as of 2021).
Only by chapters six and seven will even those with a decent grasp of mathematics start to come unstuck. Birch and Swinnerton-Dyer, a problem about rational solutions of elliptic curves, and the Hodge Conjecture, which I won't even attempt a precis of, are so couched in layer upon layer of abstract mathematics that the author deserves praise for even attempting a lay explanation.
A very committed reader will get something out of these chapters - but I think the author was right to hold up his hands in a couple of places and say: this might just be too hard for you. He gives an opt-out about halfway through the Birch and Swinnerton-Dyer chapter; while he advises the reader not to feel bad if they have to bail out of the whole of the Hodge Conjecture chapter. Which is not to criticise the author for lack of trying. Rather, I think he does well on two counts. One: he did manage to give me some level of understanding of these highly esoteric problems. Two: he makes clear that these are incredibly advanced and, at least in the case of Hodge, he makes clear that mathematicians are not even in agreement as to what it means.
All in all, the author deftly leads the engaged amateur through a lot of extremely difficult problems, to such an extent that I've already made use of my understanding of the P vs NP problem in my work (I occasionally have to think about complexity theory). At the same time, he doesn't shy away from the very hardest topics but also shields the reader from the very worst of it and attempts to keep up their confidence with welcome honesty about his own struggles with the mathematics of the last few chapters. Overall, I doubt a better primer on the Millennium Problems exists and am thoroughly glad I finally got around to reading this.
January 18, 2015
This book is an introduction to the 7 math problems designated by the Clay Mathematical Institute as "Millenium" problems and carry an award of 1 million dollars each. The problems are mathematically dense and I was able to follow in part perhaps that first four descriptions. The 5th and 6th were tough (I basically simply read through the 6th) and the 7th was completely impossible to understand. Though I think it is the nature of these problems and not a shortcoming in the author. But still it was good quick read.
January 23, 2021
Mathematics is both challenging and confusing for some. Well, there are seven ( actually six) problems that are unsolved by any mathematician. A Holy Grail, one could say. Conquering one of these problems would make you famous, but they are challenging for a reason.
The seven problems are as follows; The Riemann Hypothesis, The Yang-Mills Theory, The P versus NP problem, The Navier-Stokes Equations, The Poincare Conjecture, The Birch and Swinnerton-Dyer Conjecture, and The Hodge Conjecture. Of the seven, only the solution to the Poincare Conjecture is available.
"The Millennium Problems" educates the layman on the particulars of each problem without going too deep into the mathematics. Devlin discusses the different applications that a solution would bring.
Finally, some of the problems have an appendix. The Riemann Hypothesis section has three appendices that discuss how we know there is an infinite number of primes and other such results. The book doesn't limit itself to mathematics. During the exploration of the Yang-Mills Theory, we get into physics.
Keith Devlin does an excellent job of explaining why these problems are significant and hard to solve.
The seven problems are as follows; The Riemann Hypothesis, The Yang-Mills Theory, The P versus NP problem, The Navier-Stokes Equations, The Poincare Conjecture, The Birch and Swinnerton-Dyer Conjecture, and The Hodge Conjecture. Of the seven, only the solution to the Poincare Conjecture is available.
"The Millennium Problems" educates the layman on the particulars of each problem without going too deep into the mathematics. Devlin discusses the different applications that a solution would bring.
Finally, some of the problems have an appendix. The Riemann Hypothesis section has three appendices that discuss how we know there is an infinite number of primes and other such results. The book doesn't limit itself to mathematics. During the exploration of the Yang-Mills Theory, we get into physics.
Keith Devlin does an excellent job of explaining why these problems are significant and hard to solve.
August 21, 2020
Whew, exceptionally difficult. The Millennium Problems remain the most difficult and important unsolved mathematical problems of the modern day (with the exception of the Poincare Conjecture, which was proved). Devlin does a nice job breaking down some absurdly abstract mathematics, although my math degree still had to do a lot of heavy lifting (and I still struggled with a a lot of the math). This probably isn't the best introduction for someone with no math background, but I am content with the knowledge that I can at least give a general explanation of each of the seven problems.
January 16, 2024
During the last chapter on the hodge conjecture I started recognizing terms that my professor taught us about. I could probably solve it in a few weeks.
December 12, 2007
At the beginning of the twentieth century, David Hilbert gave a talk in which he posed 23 problems in mathematics. The solutions -- or attempted solutions -- to these problems became a major part of the story of mathematics in the twentieth century. At the start of the twenty-first century, the Clay Math Institute posed seven problems to serve as a similar challenge for another century of mathematical research. Keith Devlin, a writer of popular mathematics and the "math guy" on NPR's "All Things Considered," undertook the challenge of describing these challenge problems to a general audience.
The problem with this book is that Devlin tries to explain this cutting edge of mathematics to a really general audience. The explanation is too great a challenge. Devlin tries to lay out the basic setting of each of the problems, to discuss a little of the origins of each problem and why each problem is hard and interesting. But some of these problems are hard to even describe, and too often Devlin is reduced to saying that the problems are hard, important, and interesting without convincingly showing that the problems are hard, important, and interesting.
I liked the book. I don't think I could do better. And still I was disappointed at the end.
The problem with this book is that Devlin tries to explain this cutting edge of mathematics to a really general audience. The explanation is too great a challenge. Devlin tries to lay out the basic setting of each of the problems, to discuss a little of the origins of each problem and why each problem is hard and interesting. But some of these problems are hard to even describe, and too often Devlin is reduced to saying that the problems are hard, important, and interesting without convincingly showing that the problems are hard, important, and interesting.
I liked the book. I don't think I could do better. And still I was disappointed at the end.
September 6, 2014
A whirlwind tour of seven of the most important and difficult unsolved problems in mathematics today (actually, one of them, the Poincaré Conjecture, has been solved since the book was written).
As an introduction, I think the book does a good job of imparting the general idea of each of the problems, even the Hodge Conjecture which in its full glory is well past my ability to comprehend without a few years of study. I recommend this book as a good starting point for getting an overall understanding of these problems.
There are other accessible books devoted to at least some of the problems and they would be good follow-on reading. For the Poincaré Conjecture try _Poincaré's Prize_ (George Szpiro) and for the Riemann Hypothesis try _Prime Obsession_ (John Derbyshire).
As an introduction, I think the book does a good job of imparting the general idea of each of the problems, even the Hodge Conjecture which in its full glory is well past my ability to comprehend without a few years of study. I recommend this book as a good starting point for getting an overall understanding of these problems.
There are other accessible books devoted to at least some of the problems and they would be good follow-on reading. For the Poincaré Conjecture try _Poincaré's Prize_ (George Szpiro) and for the Riemann Hypothesis try _Prime Obsession_ (John Derbyshire).
March 24, 2007
Yeah, you just TRY explaining superalgebras to nonspecialists in a small paperback book! I give them two stars for effort, but overall this book's a wash.
September 14, 2017
I have included in my review a couple of quotes and my comments on them. Much of the book is very hard to understand for a non-mathematician. He does give some insight into his worldview as the following quotes demonstrate.
Pages 179 & 180
"In four dimensions the bottle would not have to pass through itself. To the person in the street, an object that exists in four-dimensional space doesn’t really exist of course, but this trivial objection does not deter the mathematician. After all everyone “knows” that negative numbers do not have square roots, but that did not prevent mathematicians from developing the complex numbers and, moreover, using them in practical applications. (He does not mention any practical applications) Much of the immense power of mathematics comes from the fact that we can use it to investigate objects that are beyond our conception in living creatures in a three-dimensional world." (Are they investigating real objects or imaginary ones? What is the value (power?) in investigating imaginary objects which do not exist.)
page 184
"There is no mathematical reason to stop at three dimensions. You can write down equations that determine manifolds of 3, 4,5, 6, or any number of dimensions. Once again, these considerations turn out to be more than idle speculation. The mathematical theories of matter that physicists are currently working on view the universe we live in as having 11 dimensions. According to these theories, we are directly aware of three of those dimensions, and the others manifest themselves as various physical features such as electromagnetic radiation and the forces that hold atoms together." (We are only able to be aware of three dimensions but because mathematicians can write equations using any number of dimensions, we are to somehow accept them as real? When authors invent things in science stories we call that science fiction. Perhaps we should call the same thing done in math “math fiction”!)
Pages 179 & 180
"In four dimensions the bottle would not have to pass through itself. To the person in the street, an object that exists in four-dimensional space doesn’t really exist of course, but this trivial objection does not deter the mathematician. After all everyone “knows” that negative numbers do not have square roots, but that did not prevent mathematicians from developing the complex numbers and, moreover, using them in practical applications. (He does not mention any practical applications) Much of the immense power of mathematics comes from the fact that we can use it to investigate objects that are beyond our conception in living creatures in a three-dimensional world." (Are they investigating real objects or imaginary ones? What is the value (power?) in investigating imaginary objects which do not exist.)
page 184
"There is no mathematical reason to stop at three dimensions. You can write down equations that determine manifolds of 3, 4,5, 6, or any number of dimensions. Once again, these considerations turn out to be more than idle speculation. The mathematical theories of matter that physicists are currently working on view the universe we live in as having 11 dimensions. According to these theories, we are directly aware of three of those dimensions, and the others manifest themselves as various physical features such as electromagnetic radiation and the forces that hold atoms together." (We are only able to be aware of three dimensions but because mathematicians can write equations using any number of dimensions, we are to somehow accept them as real? When authors invent things in science stories we call that science fiction. Perhaps we should call the same thing done in math “math fiction”!)
July 25, 2025
I usually reserve 1-star reviews for Michio Kaku's books, but this book is incredibly poorly researched and proofread. Avoid.
Well, this is not encouraging. Page 5 and already a serious error:
"...“Mass Gap Hypothesis,” which concerns supposed solutions to the Yang-Mills equations. This hypothesis is accepted by most physicists, and provides an explanation of why electrons have mass."
Absolutely not! It has nothing whatsoever to do with the mass of electrons. In the particular Yang-Mills theory QCD SU(3) with no quarks, it means the glueballs would have a non-zero mass.
p.56 - Why would you spell Carl Gauss with a K, Karl Gauss? The German 10 mark banknote uses a C, his own published papers use a C.
p.79 - "For all its power, Einstein’s theory of special relativity applied only when two or more frames of reference moved relative to each other at constant velocity."
No. Wrong. The special theory can certainly handle accelerations. The difference between the special and general theories is that in SR spacetime is flat and in GR spacetime is curved.
p.80 - "In 1919, the British astronomer Sir Arthur Eddington made an accurate observation of the planet Mercury during a total solar eclipse, finding that its position in the sky was “wrong.” The explanation was that the sun caused the light from the planet to bend, resulting in the apparent shift in its position."
Utter nonsense! Mercury was not involved. Stars near the limb of the Sun were observed. Did no one proofread this book?
p.92 - "For the weak force, this difficulty was overcome in 1967 by Sheldon Glashow, Abdus Salem, and Steven Weinberg ..."
It's SALAM, not SALEM. Again, where is the proofreader?
p.92 - "The next step was taken by Howard Georgi and Glashow a decade later. Using a larger group known as SU(3) x U(2), they managed to incorporate the strong force as well, calling this
new theory quantum chromodynamics (QCD)."
Oh my God! No.
OK, you know what? I'm going to stop correcting the numerous errors in this book. The author can pay me if he wants more errata identified.
Well, this is not encouraging. Page 5 and already a serious error:
"...“Mass Gap Hypothesis,” which concerns supposed solutions to the Yang-Mills equations. This hypothesis is accepted by most physicists, and provides an explanation of why electrons have mass."
Absolutely not! It has nothing whatsoever to do with the mass of electrons. In the particular Yang-Mills theory QCD SU(3) with no quarks, it means the glueballs would have a non-zero mass.
p.56 - Why would you spell Carl Gauss with a K, Karl Gauss? The German 10 mark banknote uses a C, his own published papers use a C.
p.79 - "For all its power, Einstein’s theory of special relativity applied only when two or more frames of reference moved relative to each other at constant velocity."
No. Wrong. The special theory can certainly handle accelerations. The difference between the special and general theories is that in SR spacetime is flat and in GR spacetime is curved.
p.80 - "In 1919, the British astronomer Sir Arthur Eddington made an accurate observation of the planet Mercury during a total solar eclipse, finding that its position in the sky was “wrong.” The explanation was that the sun caused the light from the planet to bend, resulting in the apparent shift in its position."
Utter nonsense! Mercury was not involved. Stars near the limb of the Sun were observed. Did no one proofread this book?
p.92 - "For the weak force, this difficulty was overcome in 1967 by Sheldon Glashow, Abdus Salem, and Steven Weinberg ..."
It's SALAM, not SALEM. Again, where is the proofreader?
p.92 - "The next step was taken by Howard Georgi and Glashow a decade later. Using a larger group known as SU(3) x U(2), they managed to incorporate the strong force as well, calling this
new theory quantum chromodynamics (QCD)."
Oh my God! No.
OK, you know what? I'm going to stop correcting the numerous errors in this book. The author can pay me if he wants more errata identified.
May 25, 2018
It is a book that attempts to explain the 7 Millennium problems ( which can be called "extremely difficult unsolved mathematical problems for the Humankind") by using more words and less mathematics. Out of those 7 problems, fortunately, One (The Poincare Conjecture) has been solved by Grigori Perelman(who rejected the $1 million prize) in 2006.
Did I said "$1 million"? Yes, apart from achieving the satisfaction of solving an "unsolved" math problem, Clay Mathematics Institute offers $1 million prize to the solver(s).
This book starts with the basics, introduces the history, and formulate the problems in a layman language. If you just want to know where the frontier of modern mathematics currently lie, this book could be your possible start. However, if you hold or are working towards a (post)graduate degree in Mathematics, you may prefer to visit the website( http://www.claymath.org/millennium-pr... ) directly.
Did I said "$1 million"? Yes, apart from achieving the satisfaction of solving an "unsolved" math problem, Clay Mathematics Institute offers $1 million prize to the solver(s).
This book starts with the basics, introduces the history, and formulate the problems in a layman language. If you just want to know where the frontier of modern mathematics currently lie, this book could be your possible start. However, if you hold or are working towards a (post)graduate degree in Mathematics, you may prefer to visit the website( http://www.claymath.org/millennium-pr... ) directly.
March 13, 2024
Almost-comically dense/unapproachable material.
Truly reveals how abstract and just downright - forget having a crack at these problems, forget understanding these problems, even forget understanding the sequence of actions required to begin to understand what the problems actually are - unfathomable contemporary mathematics is. Consequence of the move away from maths as description of, and necessary grounding in, reality and real things that we can see and touch and calculate to taking the axioms and logic as far as we possibly can. Sort discovered stuff to invented stuff. Although yet somehow (as the millenium problems illuminate) these mutated abstractions seem to always come in handy. Strange.
Truly reveals how abstract and just downright - forget having a crack at these problems, forget understanding these problems, even forget understanding the sequence of actions required to begin to understand what the problems actually are - unfathomable contemporary mathematics is. Consequence of the move away from maths as description of, and necessary grounding in, reality and real things that we can see and touch and calculate to taking the axioms and logic as far as we possibly can. Sort discovered stuff to invented stuff. Although yet somehow (as the millenium problems illuminate) these mutated abstractions seem to always come in handy. Strange.
February 14, 2021
I thought that this was an excellent overview of the problems, made accessible to people who have a keen interest in Mathematics. The book does an excellent job of describing how the topics that the problems use were motivated by the real world, and also showcases what the large variety of topics Mathematics deals with.
This entire review has been hidden because of spoilers.
March 1, 2021
A brilliant description of the dense field of mathematics on complex subjects and of the 7 problems at the frontier of the human mind in today's century. Read it to understand the subject more than anything. Especially the history of the subject.
August 8, 2018
I don't think the author is a very good explainer.
November 3, 2018
The last two chapters shattered my delusion of being an amateur mathematician but the first five were pretty dope.
May 8, 2019
Simplistic enough to attract many but complex enough to inspire the passionate mathematician.
October 29, 2024
Fantastic read! Loved every page.
March 6, 2023
Phew...A bit of a difficult book to go through (because of how lost I was most of the time). But, I would like to applaud the author on taking up this almost impossible task of explaining the seven most unsolved problems in the mathematics world to an ordinary audience...an excellent job. For me, the clearest explanations definitely were the Riemann Hypothesis and the P vs NP problem. I learnt a lot through this book :)
August 7, 2018
Nice enjoyable survey.
Even managed to make things The Birch and Swynerton-Dyer conjecture seem somewhat understandable.
Not so much the Hodge conjecture though.
Even managed to make things The Birch and Swynerton-Dyer conjecture seem somewhat understandable.
Not so much the Hodge conjecture though.
October 2, 2016
Quite good summaries of complex mathematical problems. It was a pleasant read.
June 12, 2010
Attention, comme vous pouvez le deviner au titre, ceci n'est pas un roman !
Il n'est pas, pour autant, à mettre entre toutes les mains... :-)
Ce livre s'adresse aux matheux, ou au minimums aux personnes non allergiques aux maths, ayant une certaine curiosité envers ce sujet...
L'auteur réussi un exploit : s'il attend du lecteur un minimum de culture mathématique (niveau lycée), il rappelle rapidement les concepts de base (ce qu'est un logarithme ou un sinus) pour rafraîchir la mémoire et construire sur ses bases des concepts plus élaborés.
En même temps, il parvient à garder vivant ce sujet un peu aride en donnant des détails bibliographiques des grands mathématiciens ayant fait avancer la science et mené aux énigmes invoqués.
Les dites énigmes sont des problèmes extrêmement difficiles, peut-être même impossibles à résoudre. Elles sont basées sur des hypothèses, assertions, formules, devinées par des génies, vérifiées en pratique, mais que personne n'a pu prouver de façon formelle qu'elles sont justes (ou fausses en certains cas très particuliers non encore trouvés...).
Un peu comme le théorème de Fermat ou des quatre couleurs, invoqués dans le livre mais n'y figurant pas car ils ont finalement été démontrés après que des centaines de mathématiciens s'y sont cassé les dents pendant des années...
Donc, même si de nombreux concepts me sont passé largement au-dessus de ma tête (et l'auteur avoue ne pas avoir tout compris lui-même alors que c'est un mathématicien professionnel expérimenté !), le livre reste très plaisant et très didactique.
Il n'est pas, pour autant, à mettre entre toutes les mains... :-)
Ce livre s'adresse aux matheux, ou au minimums aux personnes non allergiques aux maths, ayant une certaine curiosité envers ce sujet...
L'auteur réussi un exploit : s'il attend du lecteur un minimum de culture mathématique (niveau lycée), il rappelle rapidement les concepts de base (ce qu'est un logarithme ou un sinus) pour rafraîchir la mémoire et construire sur ses bases des concepts plus élaborés.
En même temps, il parvient à garder vivant ce sujet un peu aride en donnant des détails bibliographiques des grands mathématiciens ayant fait avancer la science et mené aux énigmes invoqués.
Les dites énigmes sont des problèmes extrêmement difficiles, peut-être même impossibles à résoudre. Elles sont basées sur des hypothèses, assertions, formules, devinées par des génies, vérifiées en pratique, mais que personne n'a pu prouver de façon formelle qu'elles sont justes (ou fausses en certains cas très particuliers non encore trouvés...).
Un peu comme le théorème de Fermat ou des quatre couleurs, invoqués dans le livre mais n'y figurant pas car ils ont finalement été démontrés après que des centaines de mathématiciens s'y sont cassé les dents pendant des années...
Donc, même si de nombreux concepts me sont passé largement au-dessus de ma tête (et l'auteur avoue ne pas avoir tout compris lui-même alors que c'est un mathématicien professionnel expérimenté !), le livre reste très plaisant et très didactique.
January 1, 2016
The Millennium Problems are seven problems in mathematics identified by the Clay Mathematics Institute as being particularly difficult and important. A one million dollar prize is offered for the solution of each problem. The awards have something of a precursor going back to 1900 when David Hilbert posed 23 problems of importance to mathematics. Almost all the Hilbert problems have been solved. Only one problem from Hilbert's list makes it onto the Millennium Problems: The Riemann Hypothesis.
The author does a good job of trying to make the problems understandable to the layman. But it's a difficult task. The evolution of mathematics and physics forces the cutting edge to be further and further removed from intuition and experience. The first five problems are not too bad. I could get the overall picture. But by problem six, it was starting to read like Greek. Problem seven is so difficult to explain that the author spends several paragraphs explaining and apologizing for how difficult the problem is to explain. He is not kidding. I would have done just as well to have read the sentences backward.
The author does a good job of trying to make the problems understandable to the layman. But it's a difficult task. The evolution of mathematics and physics forces the cutting edge to be further and further removed from intuition and experience. The first five problems are not too bad. I could get the overall picture. But by problem six, it was starting to read like Greek. Problem seven is so difficult to explain that the author spends several paragraphs explaining and apologizing for how difficult the problem is to explain. He is not kidding. I would have done just as well to have read the sentences backward.
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October 16, 2012La maniera scelta da Devlin per spiegare i problemi mi piace molto: partendo con quello che appare come un approccio esageratamente vago, arrivando a volte a tirare in ballo anche i Greci ed il teorema di Pitagora, in pochissime pagine riesce a restringere il campo e vi ritrovate immersi nelle congetture e nelle ipotesi matematiche più difficili del giorno d'oggi, essendo felicemente passati attraverso un po' di Storia e di aneddoti. Inoltre, bella l'idea di una piccola bibliografia di libri per chi vuole addentrarsi pur restando un non professionista; e valida anche la postfazione del sempre bravissimo Odifreddi. (Anche se probabilmente l'idea più vincente del libro è stata ordinare i problemi per difficoltà crescente, in modo da non scoraggiare davvero il lettore almeno fin quasi alla fine.)
July 12, 2014
Hmmm, Devlin takes on a tricky task in this book, as some of the problems described are so hard to describe (let alone solve!) that he admits that even he does not understand the problem. Having said that, he does an excellent job with some of the problems, such as the Riemann Hypothesis, and the introductory material in each chapter giving the mathematical (and physics) background is very good. He also provides biographical snippets for some of the mathematicians mentionned which helps to keep the book flowing. Worth a read.
February 11, 2023
You have to appreciate Devlin's gumption in attempting to explain the hardest problems in the mathematical world to regular joes and janes, and he does a pretty good job. But there's no getting around the fact that he pretty much punts on the last two. I mean, he *sort of* tries, but he pretty much tells the reader that it's too complicated to explain. I hoped for at least a little more attempt.
October 4, 2014
Devlin tries and for that he deserves Praise. The seven millennium problems are described as simply as possible, which is still too complicated for most folks (the last two chapters sent my head spinning and were the reason I took so long to finish this!). An interesting look at complex algebra and complex systems and a fun behind the scenes glance at the histories and culture of mathematicians. Try it out but be warned, it's a challenge.
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